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Let $A$ be a finite dimensional $k$-algebra with finite global dimension. How can I prove that the category $D^b(A)$ (bounded derived category of the category of left finitely generated $A$-modules) is equivalent to $K^b(_AP)$ (bounded homotopy category of complexes made with left projective modules) ?

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I'm a bit puzzled about what is missing: Fact 2) from your other question shows that the inclusion $K^b({}_AP) \to D^b(A)$ is fully faithful and to show that it is essentially surjective (and hence an equivalence of categories) you need only prove that every bounded complex of f.g. modules is quasi-isomorphic to a bounded complex of f.g. projectives (I assume that's what you mean). Two possible proofs of that latter fact were mentioned in the comments there. Could you elaborate on what is unclear? – commenter Nov 8 '12 at 23:36

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